Examine the leaves, including the undersides, for pests. Aphids and thrips could be feeding on the curry plants, causing the leaves to start dying at the tips. Treat damaging insects as soon as possible, such as naturally with an insecticidal soap or with an appropriate insecticide.
Flush the soil with 2 to 3 inches of water if you have recently fertilized the curry plants. You may have over-fertilized. Also, rinse the leaves. Fertilizer could have touched and burned them.
Remove an affected leaf and take it to a nursery or county extension office for inspection and diagnosis. The browning could be caused from a disease, such as bacterial leaf spot. If it is a disease, take steps to reduce its spread. For example, remove and discard the affected leaves, lower the humidity around the plants or treat the curry plants with a fungicide. Treatment varies depending on the specific disease.